Absolutely closed semigroups.

Let C be a class of topological semigroups. A semigroup X is called absolutely C -closed if for any homomorphism h : X → Y to a topological semigroup Y ∈ C , the image h [ X ] is closed in Y . Let T 1 S , T 2 S , and T z S be the classes of T 1 , Hausdorff, and Tychonoff zero-dimensional topological semigroups, respectively. We prove that a commutative semigroup X is absolutely T z S -closed if and only if X is absolutely T 2 S -closed if and only if X is chain-finite, bounded, group-finite and Clifford + finite. On the other hand, a commutative semigroup X is absolutely T 1 S -closed if and only if X is finite. Also, for a given absolutely C -closed semigroup X we detect absolutely C -closed subsemigroups in the center of X .